No, the UPS doesn't concern what you have inside the machine, it's more connected to the Max Power of the PSU.....
If your rig is underpowered , then the slightest fluctuation in the mains is going to trip the UPS causing a instant shutdown.
If your PSU matches up to the UPS then you get a 5 to 15 minute power backup while using max power to be safely out of the red zone and shut your PC down.
If the PSU's power intake is more than what the UPS can supply then again you have a shutdown when the power goes out plus, you will kill your battery in the UPS real fast.....
Your 450Watt PSU , it seems was sufficient enough for the 1055 and 2X 5870CF....
which is pretty credible...... but usually doesn't happen.....
For your 450W PSU the 0.5KVA UPS is enough, for a 850WPSU you'll need a 1KVA UPS....
Simple, the UPS powers the PSU and the PSU powers the Rig.
Thus if you have low power coming into the PSU the system is going to reboot if your UPS is not capable of handling the PSU power needs it's going to trip
it's called overload.....
My system here has that since it's connected to an Inverter, so the inverter tells me it's overloaded when I have a power cut and need to shutdown 2 out of three of my rigs.....